5 - Existence and Uniqueness of Solutions to Differential Equations

Solutions in this case refers to particular solutions, as in, solutions for a differential equation at a point.

Consider the example general form of a differential equation: This means, the derivative of y with respect to x, that is a function in terms of x and y. A function being in terms of something means that it has those variables in the equation.

We are talking derivatives here. There are only 3 cases that can happen:

1 - No solution at that point
2 - Infinite solutions at that point
3 - One unique solution in a neighborhood around that point.

Why neighborhood? Because we need it to be continuous. You can't have continuity at an endpoint, you need to approach it and leave it. If you have an endpoint or something else like a point that's undefined or discontinuous, a solution at that point can't exist.

You can interpret this in three equivalent statements:

  • There are no solutions at endpoints or discontinuities
  • If the function is continuous around a point there exists at least one solution to this differential equation
  • If a point is a discontinuity of the function, it can't be a solution.

Checking for uniqueness

To check if such solution is unique, we need to do something called partial differentiation.
Instead of using the d symbol, you use the fancy greek delta. Means, the partial derivative of y with respect to x.
It sounds scary but it's really not.

The partial derivative means you take the derivative of a function, but all other variables are considered constant.

So, you would treat x as you would any constant number.

Example: Treat x as a constant! Differentiate for y.

So, going back to where we were, if the partial derivative of this f(x,y) is also continuous at that point, that solution(provided it exists) is unique!

Good job.

6 - Separable Differential Equations

Interactive Graph